CC BY
12
Deformation of stressed rocks during earthquakes, rock bursts and explosions
Chen Yeqing, Gong Gupei, Xie Dongsheng
CHEN YEQING, GONG GUPEI, XIE DONGSHENG, Beijing Canbao Architecture Design & Research Institute, Beijing 100036, China
Dynamic interaction mechanism between rocks and underground arches
A procedure is presented to analyze underground protective structures subject to conventional weapons effects. A radiation damping term is added to a simple mass-resistance system to consider structure-soil interaction effects. To reflect the influence of the radius to the dynamic interaction, traditional Constantino model will be applied. Based on the virtual work method, the displacement and the reflective coefficient of the arch were deduced. Combined with the distribution of the free-field of the explosion on the surface of the arch, dynamic loading of the arch can be calculated. The predicted result is also checked by numerical simulations. SUBJECT: Analysis techniques and design methods.
Key words: dynamics, tunneling.
1. Introduction
Soil-structure interaction (SSI) is important in designing underground protective structures. Several methods have been suggested to design underground structures including the effect of SSI (Wong, 1983).
In Modified Parameter Method (MPM), the dynamic characteristics of the structure are altered due to SSI effects. It is suggested the effective mass of the structure for SSI design should be increased by an appropriate amount. For rocks, validity of this method should be discussed. Engineers always determined the dynamic loading by the falling velocity of broken rocks.
In Reflected Pressure Method (RPM), the dynamic loading on the structure is divided into two parts: the free-field pressure; and the interface reflected pressure due to the velocity difference between the structure and soil. The loading on the structure is simply the free-field pressure amplified by a constant factor.
In engineering, calculation methods of dynamic loads are always determined by tests and experiences due to the complex of interactions under blast. Tested results are also restricted by experimental conditions, such as weapons and structure styles. For example, composite lines with larger spans are not concerned in the past tests. Present methods are not accurate for modern protective tunnels. Additionally, dynamic loads of arches are always modified from loads of straight panels. This modification is absent of mechanical mechanism. Radius of the arch greatly influences the interaction between structure and medium.
2. Limitation OF PRESENT methods
Present methods used to calculate dynamic loads on structures under blast loads were discussed. Variations of dynamic loads with spans were deduced from the published equations. It is found that in some equations dynamic loads are irrelevant to spans and for others dynamic loads alter only when the span is smaller than 10 m.
In summary, present methods are not suitable to calculate dynamic loads of underground protective arches with spans larger than 10 m.
© Chen Yeqing, Gong Gupei, Xie Dongsheng, 2014
b"
g 0.50 b*
-40 -20 0 20 40
9 (degree)
Fig. 1. Dynamic loading distribution
3. Distribution of the free-field
The distribution of the free-field of the explosion on the surface of the arch was deduced according to free stress field suggested by TM855-1. A time-dependent triangular style loading was found for the buried arch as shown in Fig. 1. The resulted distribution of loading is close to the numerical analysis as shown in Fig. 2.
The calculation reveals the non-uniform distribution of dynamic loads on the arch.
1.0 0.8 0.6 0.4 0.2 0.0
LS-DYNA simulation formula calculation
-20 -15 -10 -5 0 5 10 15 20
9( o)
Fig. 2. LS-DYNA simulation result and calculated free field stress for tunnels with a span of 60 m
and standoff a distance of 6.5 m
4. Strructure-medium Inteaction model
The normal interface pressures were computed by transforming the free-field stresses and ground motions to the local coordinates of the structural element and applying the pcv medium-structure interaction term, as fol1ows: [2-5]
an =af +pcvm , an > 0, (1)
in which
' Cl f > 0
c = •
\Cu if < 0
(2)
1.25
0.00
and Gn is the normal interface pressure, af is the free-field pressure normal to structural element, p is the soil density, c is the loading wave speed of soil, ca is the unloading wave speed of soil, and vm is
the relative normal velocity of structure and soil. This model has been adopted in the standard of US army (TM5-855-1, 1986).
Constantino assumed an interaction formula that provides the normal stress at the interface of the soil and structure as (Constantino & Vey 1969, Miller& Constantino 1970 )
an=af + KAW + SAW, (3)
where K is the foundation modulus related to the radius of the arch, S is the damping coefficient, AW is the relative radial displacement and AW is the relative radial velocity. This model considered the influence of the radius of the arch in dynamic interaction model.
Constantino model was adopted and developed to study the dynamic interaction mechanism and the dynamic loading distribution of underground arches. The model is expressed as (Zhang et al., in press)
E
an = 2&f -PcUi +TcL (wfi - u)■. f 2r f
(4)
where u and w denote the soil displacement normal to the interface in soils in free filed and structures, respectively. Symbol Ec is the modulus of the soil and r is the radius of the arch.
The free field stress and displacement are suggested by the standard of US army (TM5-855-1,
1986).
5. Dynamic load
The improved Constantino model will be applied to calculate the reflective coefficient of the dynamic loading combined with the energy method. A shape function of displacement of the arch will be assumed. According to the virtual work principle, the virtual work done by an equilibrium system of forced at the virtual displacement field must be equal to zero. The equation of the virtual work done by the forces loaded on the arch will be delineated to get the kinematical equation. The solution of the displacement will be deduced. And then the reflective coefficient can be calculated.
Assuming a virtual displacement at the vault of the arch and a displacement distribution function X(p), normal displacement of the arch is SX(p). According to the virtual work principle, equation of motion of the arch is (Zhang et al., in press)
(5)
J an8X(p)ls - J qSX(p)ls - J mU0X(<p)d¥(p)ds = 0,
where m is the mass of the arch with unit arc and q is the resistance of the arch.
Finally, the dynamic loading of the arch will be achieved based on the multiplication of the free-filed and the reflective coefficient.
Omitting the detailed deductions, the average peak value of the dynamic load on the arch is given by (Zhang et al., in press)
Pi = K (aoCf) (6)
with
f
K = K
K = 2 -
1 K 2 1 + —
K
A
(1 + K3)
4N M
1
1 - e
fN sin MV1 - N2 ^
-NM
N1
4T-
sMa/i-
v+ cos M V1 - N j
(7)
(8)
s
FEFU: SCHOOL of ENGINEERING BULLETIN. 2014. N 3 (20)
à', = LN
2 5 M
and
-1 + 2
N
M (
- e
-NM
(2N2 — 1) sin W1 - N2
MV1 - N2
+ 2 N cos MV1—N2
. M
^ = Ec
4r <«Cf
where
M = atr,
N = ■
2m a
L = H,
r
and
n2 Ec a =
2mr
(9)
(10)
(11) (12)
(13)
(14)
Symbol H is the stand-off distance of the explosion. Symbols w0 and <j0 denote the peak value of displacement and stress in the free field. Symbols Cw and Cy denote the uniform distribution coefficients
of the displacement and stress at the arch surface in the free field. Symbol tr is the duration of the blast wave suggested by the standard of US army (TM5-855-1, 1986).
6. Discussions
2.0
1.8
1.6
1.4
1.2
1.0
0.0 0.2 0.4 0.6 0.8 1.0
M
Fig. 3. Reflection coefficient at l = « (Zhang et al., in press)
À is the reflective coefficient for straight panel when r , as shown in Fig. 3. Value of à increases with smaller N and M .
À2 is minus with absolute value smaller than 0.1À as shown in Fig.4, which has limited contribution to the reflective coefficient.
M
Fig. 4. Ratio of absolute value of K2 to K for N = 0.5 (Zhang et al., in press)
According to the standard of US army (TM5-855-1, 1986), K3 can be simplified as
C
K = 0.781 C^L , (15)
Cf
which is positive and its value can be close to 1.0. Assuming the displacement function is cos(^), values of C / Cf must be 0.52 and 0.97 for l = 2 and L = 1, which lead 0.81 and 0.76 for K3, respectively.
The three coefficients could make the value of K greater than 2.0.
7. Conclusions
Conclusions would be made from above discussion as follows:
(1) Present methods can not be applied to determine the dynamic load of underground protective arches with spans larger than 10 m.
(2) The distribution of the dynamic load of the arch under blast load is not uniform. On the contrary, it has a shape of triangle.
(3) For underground arches, radius greatly influences the soil-structure interaction. Dynamic load is enhanced by the radius effect.
(4) A new equation has been suggested to calculate the dynamic load of underground protective arches. Three parameters are included in the equation. The three coefficients could make the value of K greater than 2.0. This conclusion is absolutely different with existing calculation methods where K should be always smaller than 2.0.
Acknowledgements
Supports of National Natural Science Foundations of China (51078351, 51021001), Fund of Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering of Hohai University (GH200905), Fund of Aseismic Engineering Technology Key Laboratory of Sichuan Province are gratefully acknowledged.
REFERENCES
1. Constantino C.J., Vey E. Response of buried tunnels encased in foam. Proceeding of the American Society of Civil Engineers. Journal of the Soil Mechanics and Foundation Division, 1969:1159-1179.
2. Miller C.A., Constantino C.J. Structure-foundation interaction of a nuclear power plant with a seismic disturbance. Nuclear Engineering and Design, 1970;14:332-342.
3. TM5-855-1. Fundamentals of Protective Design for Conventional Weapons, US Army Engineer Waterways Experiment Station, Vicksburg, MS, 1986.
4. Wong F.S., Weidlinger P. Design of underground protective structures. ASCE Journal of Structural Engineering, 1983;109:1972-1979.
5. Zhang Y., Fan H.L., Jin F.N. In press. Study of reflective coefficients of underground protective arches under blast loads (In Chinese).
